Rust-coloured stain coming through wall

[Pete77]

Daughter has just moved into a flat and found this brick/rust-coloured stain creeping up kitchen corner that backs onto garden area.
I assume it is moisture coming through brick work with faulty pointing, but could be completely wrong.

1. Is there a proper name for this condition?
2. Should landlord get a professional in to deal with, or
3. Can it just be coated over with special liquid and, if so, what type?

Any diagnosis / answers much appreciated!

 

[opps]

The galvanised angle bead is rusting.

 

[opps]

There is clearly damp running along that angle corner. I can't see any paint bubbling though to either side of it.

 

[foxhole]

External photos? Entire wall and roof